Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes

Abstract

Abstract. The goal of this article is to analyze the observability properties for a space semi-discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuniform meshes. More precisely, we prove that observability properties hold uniformly with respect to the mesh-size under some assumptions, which, roughly, measures the lack of uniformity of the meshes, thus extending the work [Castro and Micu, Numer. Math. 102 (2006) 413-462] to nonuniform meshes. Our results are based on a precise description of the spectrum of the discrete approximation schemes on nonuniform meshes, and the use of Ingham's inequality. We also mention applications to the boundary null controllability of the 1d wave equation, and to stabilization properties for the 1d wave equation. We finally present some applications for the corresponding fully discrete schemes, based on recent articles by the author.Mathematics Subject Classification. 35L05, 35P20, 47A75, 93B05, 93B07, 93B60, 93D15.

Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis

“…Actually, this phenomenon was already observed in the work [7] corresponding to the case of a space discretization based on mixed finite elements. One can furthermore check experimentally that the dispersion diagram presents a significant gap when the domain (0, 1) is discretized using the mesh (1.3) corresponding to g θ in (1.8), while the gap vanishes when the mesh is uniform.…”

“…As the mesh is non-uniform, this seems very intricate. In fact, to our knowledge, the only case in which a spectral gap was proven for non-uniform meshes corresponds to the case of the 1d wave equation (1.1) discretized in space using the mixed finite element method, see [7].…”

Numerical approximation schemes for multi-dimensional wave equations in asymmetric spaces

“…This method also works for the finite element and mixed finite element discretization but it requires to have uniform time discretizations. Here, we can provide an estimate using the discrete version of the Rellich multipliers which applies for the full space-time discretized equations with non-uniform meshes (see [8]). …”

Resolvent estimates in controllability theory and applications to the discrete wave equation

“…Another result in this direction is presented in [10], in the context of the 1d wave equation discretized using a mixed finite element method as in [2,6]. In [10], it is proved that observability properties for schemes derived from a mixed finite element method hold uniformly within a large class of nonuniform meshes.…”

“…In [10], it is proved that observability properties for schemes derived from a mixed finite element method hold uniformly within a large class of nonuniform meshes.…”